You know that your object covers half of the height of the tower in 2 seconds. Assuming that this distance is actually the first half of the height of the tower, you can write, taking into consideration the fact that the initial velocity of the object is zero,
h_1 = underbrace(v_0)_(color(blue)("=0")) * t_1 + 1/2 g * t_1^2
h_1 = 1/2 g * t_1^2, where
h_1 - the first half of the distance;
t_1 - the time it needed to cover this distance, in your case t_1="2 s"
The total distance the object travelled can be written as
H = underbrace(v_0)_(color(blue)("=0")) * t_T + 1/2 * g * t_T^2, where
t_T - the toal time the object took to reach ground level.
You know that
H = h_"first half" + h_"second half"
H = 2 * h_"first half" = 2 * h_1
This means that you have
2 * h_1 = 1/2 * g * t_T^2
cancel(2) * 1/cancel(2) * cancel(g) * t_1^2 = 1/2 * cancel(g) * t_T^2
This is equivalent to
t_T^2 = 2 * t_1^2 = 2 * 2^2 = 8
Thus,
t_T = sqrt(8) = color(green)(2sqrt(2))
